Bridging Dimensions: Confident Reachability for High-Dimensional Controllers

TL;DR

The paper tackles verifying end-to-end controllers built from high-dimensional inputs by approximating the black-box high-dimensional controller with multiple verifiable low-dimensional controllers trained via verification-aware knowledge distillation. It then derives high-confidence safety guarantees by quantifying and conformally bounding the discrepancy between high- and low-dimensional controllers and inflating the low-dimensional reachability tubes accordingly, using both trajectory- and action-based discrepancy models. The authors implement this five-step pipeline, validate it on OpenAI Gym benchmarks (inverted pendulum, mountain car, cartpole) with DDPG-based HDCs and feedforward LDCs, and show that trajectory-based, multi-LDC inflation typically yields stronger, less conservative guarantees than single-LDC or pure conformal baselines. The approach enables scalable, probabilistic verification of complex perception-based controllers with thousands of inputs, potentially broadening the safe deployment of learning-enabled autonomous systems.

Abstract

Autonomous systems are increasingly implemented using end-to-end learning-based controllers. Such controllers make decisions that are executed on the real system, with images as one of the primary sensing modalities. Deep neural networks form a fundamental building block of such controllers. Unfortunately, the existing neural-network verification tools do not scale to inputs with thousands of dimensions -- especially when the individual inputs (such as pixels) are devoid of clear physical meaning. This paper takes a step towards connecting exhaustive closed-loop verification with high-dimensional controllers. Our key insight is that the behavior of a high-dimensional controller can be approximated with several low-dimensional controllers. To balance the approximation accuracy and verifiability of our low-dimensional controllers, we leverage the latest verification-aware knowledge distillation. Then, we inflate low-dimensional reachability results with statistical approximation errors, yielding a high-confidence reachability guarantee for the high-dimensional controller. We investigate two inflation techniques -- based on trajectories and control actions -- both of which show convincing performance in three OpenAI gym benchmarks.

Figures (4)

• Figure 1: Our verification approach for systems with high-dimensional controllers.
• Figure 2: Reachability examples of the action-based approach on the inverted pendulum. Left: safe, right: unsafe.
• Figure 3: Three phases of our approach: training, verification, and control. Orange shows high-dimensional elements, and green shows novel contributions.
• Figure 4: Training an LDC with supervision from an HDC.

Theorems & Definitions (29)

• definition thmcounterdefinition: Reachable set
• definition thmcounterdefinition: Reachable tube
• lemma thmcounterlemma
• definition thmcounterdefinition: Statistical trajectory-based discrepancy
• definition thmcounterdefinition: Statistical action-based discrepancy
• definition thmcounterdefinition: Trajectory-inflated reachable set
• definition thmcounterdefinition: Trajectory-inflated reachable tube
• theorem thmcountertheorem: Confident trajectory-based overapproximation
• proof
• definition thmcounterdefinition: Action-inflated reachable set